1. Field of the Invention
The present invention relates generally to financial asset risk analysis and more particularly to a method and apparatus for optimizing a portfolio of financial assets.
2. Background Art
Assessing the degree of risk of a financial asset is central to rational financial planning. Investors and analysts have an intuitive idea of what constitutes investment risk, but such risk is a difficult concept to define precisely so that it can be measured and quantified.
There are six main indicators or measures of investment risk that are commonly used and that apply to the analysis of stocks, bonds, mutual funds, exchange traded funds as well as to other financial instruments including portfolios of these assets. The indicators are alpha, beta, R-squared, the Sharpe ratio, relative volatility and the standard deviation. All of these measures depend on, in one way or another, a calculation of the standard deviation of the asset price, or a calculation of the variance of the asset price (which is the square of the standard deviation).
The standard deviation measures the deviations from the mean of the asset price; the greater the deviation from the mean, the greater the volatility of the asset's price. This is the basic rationale for using the standard deviation in calculations of the riskiness of an asset. If an asset's price stays constant, its standard deviation is zero. If the asset's price goes up or down, it deviates from its mean value and the average amount of all the deviations from the mean is approximately what the standard deviation measures. The formula for the standard deviation, σ, is:
  σ  =                              1          N                ⁢                              ∑                          i              =              1                        N                    ⁢                                    (                                                x                  i                                -                                  x                  _                                            )                        2                                .  where the xi are the asset prices (or a function of asset prices) and x is the mean of the asset prices (or of a function of the asset prices).
Alpha and beta are simply the constant (beta) and slope terms (alpha) of the linear regression of asset return against a benchmark. Alpha (α) is the measure of that part of an investment's return that is in excess of the benchmark's index return or the expected portfolio return that it is being compared to. The formula for alpha of a portfolio αp is:αp=Ri−(Rf+βp(Rm−Rf))where Ri=expected total portfolio return, Rf=risk free rate, βp=beta of the portfolio, and Rm=expected market return.
The beta coefficient is a measure of the volatility of an asset or portfolio in relation to the rest of the financial market. The formula for the beta of an asset is defined as:
      β    a    =            Cov      ⁡              (                              r            a                    ,                      r            m                          )                    Var      ⁡              (                  r          m                )            where rα=the rate of return of the asset and rm=the rate of return of the overall market to which the asset is being compared. Coy is the covariance and Var is the variance of their respective parameters.
By definition, the market itself has an underlying beta of 1.0 and individual stocks are ranked according to how much they deviate from the macro market (for simplicity purposes, the S&P 500 is usually used as a proxy for the market as a whole). A stock that goes up or down more than the market over time (i.e. more volatile) has a beta whose absolute value is above 1.0. If a stock moves less than the market, the absolute value of the stock's beta is less than 1.0. Higher-beta stocks mean greater volatility and are therefore considered to be riskier, but are in turn supposed to provide a potential for higher returns; low-beta stocks pose less risk but also lower returns.
The Sharpe Ratio, or Sharpe index, or Sharpe measure of reward-to-variability ratio is a measure of the excess return (or Risk Premium) per unit of risk in an investment asset or a trading strategy. It is defined as:
  S  =                    E        ⁡                  [                      R            -                          R              f                                ]                    σ        =                  E        ⁡                  [                      R            -                          R              f                                ]                                      var          ⁡                      [                          R              -                              R                f                                      ]                              where R=the asset return, Rf=the return of a benchmark asset, such as the risk free rate of return, E[R−Rf]=the expected value of the excess of the asset return over the benchmark return, and σ is the standard deviation of the excess return. The Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken. When comparing two assets each with the expected return E[R] against the same benchmark with the same return Rf the asset with the higher Sharpe ratio gives more return for the same risk. Investors are often advised to pick investments with higher Sharpe ratios.
The R-squared statistic measures how well one asset is correlated with another asset or the market as a whole. An R-squared value of 1.0 indicates a perfect correlation, a value of 0 indicates no correlation and a value of −1 a perfect negative correlation. The formula for R-squared of two assets X and Y is:r(X,Y)=Cov(X,Y)/σxσy where Cov{X,Y)=the covariance of X and Y and σxσy=the standard deviation of X multiplied by the standard deviation of Y.
The Relative Volatility Statistic is the ratio of the standard deviation of the asset return rate (or function of the return rate) divided by the standard deviation of the market return that the asset is being compared to. The formula for Relative Volatility of an asset is:Relative Volatility(Asset)=SD(Asset Return)/SD(Market Return)where SD(Asset Return)=the standard deviation of the asset return and SD(Market Return)=the standard deviation of the market return.
As previously noted, each of the described prior art risk measures use a calculation of the standard deviation of the asset price or a calculation of the variance of the asset price as a core component of the risk calculation. Disadvantageously, these prior art risk measures do not adequately describe two essential components of risk of a financial asset; the range of total return component and an asset hold time component. Consequently the range and variability of total returns of an asset and the likelihood that returns from the held asset will be available at a specific time are not provided by these risk measures.
By way of illustration, the total return 100 of an asset X is graphed as it changes over time in FIG. 1. The total return 200 of an asset Y is graphed as it changes over time in FIG. 2. The period of the graph of the total return 200 of asset Y is half that of the graph of the total return 100 of asset X. Therefore, if the total return of asset Y is in the negative total return region (below the horizontal axis), asset Y will recoup its initial value in half the time as compared to asset X. For this reason, asset Y is considered to be less risky than asset X. The mean total return for each asset is zero and coincides with the horizontal axis. However, the standard deviation for both assets is the same (0.711), thus showing that the standard deviation does not measure the frequency of change.
The standard deviation also does not take into account the direction of movement of an asset's total return and thus does not provide a measure of this intuitive notion about the riskiness of an asset's total return. With reference to FIG. 3, the total return 300 of an asset A is graphed as it changes over time. The total return 400 of an asset B is graphed as it changes over time in FIG. 4. The total return 300 of asset A starts at zero, goes up to 100%, goes back to zero, goes up to 100%, and then returns to zero. In the same time period, the total return 400 of asset B starts at zero, goes up to 100%, goes back to zero, goes down to 50%, and then return to zero. The standard deviation of both graphs is 0.548. Clearly, investing in asset B is riskier than investing in asset A because there is a substantial negative total return of 50% while the total return of asset A is never negative. The standard deviation does not distinguish between the degree of riskiness of assets A and B despite the fact that their total return volatilities are remarkably different.
By definition, the standard deviation does not distinguish between positive and negative deviations from the mean. A deviation below the mean is the same as a deviation above the mean because the deviation is squared in the standard deviation formula. However, an asset having a declining total return is generally considered riskier than an asset having an increasing total return.
The inability of the standard deviation to provide a measure of the direction of the deviations from the mean is illustrated in FIG. 5. The increasing asset price 500 of an asset C and the decreasing asset price 510 of an asset D are graphed as they change over time. Each graph has a standard deviation of 4.13 even as the price of asset C doubles and the price of asset D goes to zero.
Finally, the standard deviation is a mean-centric statistic that always underestimates the volatility of the price and total return of an asset. As the standard deviation measures approximately the average deviation from the mean, it does not measure the range or extremes of price and total return movements of an asset. Information regarding such range and extremes is of value in assessing the riskiness of an asset because investment in assets having wide extremes entails the greatest risk and potentially the greatest loss to the investor.
By way of example, and with reference to FIG. 6, the asset price 600 of an asset E is graphed as it changes over time. The graph shows three cyclical changes in the asset price 600 starting from an acquisition price of 2.0, increasing to a price of 3.0, decreasing to a price of 1.0 and then increasing to the acquisition price of 2.0. The hold time for asset E consists of three such cycles. The mean asset price for the duration of the asset hold time is 2.0. The standard deviation, since it measures deviation from the mean, will measure the greatest deviation as 1.0 from the mean 2.0. However, the greatest drawdown, from the high price of 3.0 to the low price of 1.0, is a potential loss of 2.0. If asset E is acquired at a local maximum, the potential loss is two-thirds of its value, a value greater than any deviation from the mean indicates. An investor acquiring asset E, using prior art risk measures based on the standard deviation, would have no means of knowing the volatility of the asset's price or of the magnitude of the potential loss.
Prior art risk measures based on the standard deviation thus suffer many disadvantages. Such measures do not provide complete information related to the variability of total returns of an asset or portfolio and the likelihood that returns from the asset or portfolio will be available at a specific time. Furthermore, such measures do not take into account the direction of movement of an asset's price or of a portfolio's unit price. Finally, prior art risk measures underestimate the volatility of an asset's price and of a portfolio's unit price.